What Is How Many?

Foundations

It would be natural to suppose that mathematical foundations might begin with arithmetic, or plane geometry, rather than numbers; after all, everyone knows what a number is, right? Not so fast: it can be remarkably difficult to say exactly what a number is. Go ahead, try it. If you are tempted to say, “it's what tells you how much or how many of something,” then that's a good effort. Only trouble is, that's not a definition, but a characterization. It's like saying, “a chair is something you sit on.” Useful information, no doubt, but we'd be a bit disappointed if we looked up “chair” in a dictionary and found no more than “something you sit on.” After all, one can sit on lots of things that aren't chairs.

Number proceeds from unity.

—Aristotle

The problem of “what is a number,” is an old one. To tackle it, mathematicians in the late 19th and early 20th centuries (particularly Cantor, Dedekind, Frege, Peano, Russell, and Whitehead) turned to a new (at the time) branch of mathematics called set theory. They didn’t solve the problem, but they developed a beautiful theory which can be used to model and extend our primitive (i.e., “given” or “intuitively obvious”) sense of number. A good deal of modern mathematics is now founded on this work—using, at root, nothing more sophisticated than the set operations of union and intersection with which you may already be familiar. We’ll be wanting to use these operations shortly, so let's review them now. We begin with defining the notion of a set:

A Set is any well-defined collection of objects.

In other words, any collection considered as a single thing. By “well-defined,” we mean that we can always tell when something is an element of the set in question, or when it isn't—no ambiguity. By “object” we mean absolutely anything: physical objects, ideas, colors, abstractions, and anything else you care to think of (that can form part of a well-defined collection).

We want to be able to write our sets down, and there is an established way of doing this. We first designate a symbol to stand for the set itself, usually a capital letter like A or S or some such. Then, we use (curly) braces to enclose some representation of the elements of the set, as follows:

\[A=\{\ldots\mbox{elements of }A\}\]

How do we represent the elements inside the braces? There are two ways. The first and best is simply to list them. For example, if A is a set of colors, we could write it down this way:

\[A=\{\mbox{red},\mbox{green},\mbox{blue}\}\]

Sometimes, however, listing the elements is not convenient or even possible. In that case we would use a rule method, using a statement like “all shades of green,” or even, “all colors” to represent our set.

When something is an element of a set, we denote this with a special symbol (that looks kind of like a curvaceous “E”):

\[\mbox{blue}\in A\]

This means, “blue is an element of the set A.”

Much of the power of set theory arises from the fact that we can form sets whose elements are other sets. For example, if A, B, and C are sets of colors, we could form a set of sets of colors.

Notice that the sets A, B, and C have some elements in common. For instance, A and B both contain the element “blue.” If it happens that every element of a set is also contained in some other set, then we say that the first set is a subset of the other set, and we denote this with a big “U-shape” lying on its side. For instance, if D is the set containing only “blue,” then we could write,

\[D \subset A\]

or, equivalently,

\[\{\mbox{blue}\} \subset \{\mbox{red},\mbox{green},\mbox{blue}\}\]

We consider every set to be a subset of itself. (Funny thing to do, really, but it makes sense, sort of. At least, it matches the definition of subset.) Also, there is one set that is a subset of every set, namely the empty set—the set with no elements. This is often denoted by a circle with a line through it, or a pair of braces with nothing between them.

\[\emptyset = \{\,\,\} \]

The empty set is, in a vacuous way, a subset of every set:

\[\emptyset \subset X \mbox{ for every set } X \]

Finally, we are ready for the “set operations” of union and intersection. The union of two sets A and B is the set containing all the elements that are in either A or B. Thus, if A and B are the two sets of colors above, then we have,